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Understanding how to factor binomials is crucial when simplifying algebraic expressions. A binomial is an algebraic expression that has two terms, such as \(a + b\) or \(a^2 - b^2\). Factoring binomials often comes down to recognizing patterns. One common pattern is the difference of squares, which is an expression of the form \(a^2 - b^2\). This pattern can be factored into \(a - b)(a + b)\).In our exercise, \(y^2 - 9\) is a perfect example of the difference of squares since \(9\) can be written as \(3^2\). This allows us to factor \(y^2 - 9\) into \(y - 3)(y + 3)\). Recognizing and applying these patterns is a skill that can make solving algebraic problems, such as multiplication and division, significantly more manageable.

Simplifying expressions is about making an algebraic expression as 'clean' or as straightforward as possible. This often involves a variety of actions, such as combining like terms, factoring, and canceling terms. Simplification can reduce the complexity of problems and reveal a more digestible form, which is especially useful for understanding the nature of an equation or function.

Combining Like Terms

When we talk about simplifying expressions that involve multiplication, as in our original exercise, it is essential to look for terms that can be canceled to make the expression simpler. For instance, after factoring the binomial \(y^2 - 9\), we get an expression that has a common term in the numerator and the denominator. This leads us directly to our next concept, canceling like terms.

Canceling like terms is a simplification process used when the same term appears in both the numerator and the denominator of a fraction. In algebra, if a term appears both atop and below the fraction bar and is not zero, we can 'cancel' it out. This is because dividing something by itself results in one, and multiplying or dividing by one does not change the value of the other terms.In the solution provided, once the binomial is factored, we observe that \(y - 3\) is present on both sides of the fraction. By canceling \(y - 3\) out, we are left with \(y + 3\) times \(4\), which then only needs to be multiplied together to reach the final simplified form of \(4y + 12\). This step is not just about making the math easier; it's about understanding the underlying principles that facilitate simplification in algebra and applying these principles systematically.